Regular zero locus ($\mathrm{rank}(ds{|}_V)=r$)

The zero locus of a holomorphic section of a vector bundle of a rank-$r$ bundle is regular when its derivative has rank $r$ along the locus. Its kernel is then the tangent bundle of the smooth zero set. Smoothness of the underlying set alone is insufficient: the square of a reduced defining function has the same set of zeros but zero derivative there.